Proving two functions are equivalent given equal integrals without fundamental theorem of calculus

Question

I want to prove that given two functions $f$ and $g$ that are continuous on the interval $[a,b]$, $\int_a^x f(t) \, dt=\int_a^x g(t) \, dt \; \forall x \in [a,b]$ implies that $f=g$ on the interval without using the fundamental theorem of calculus or Lebesgue integration. The conclusion seems relatively intuitive, but I don't think claiming intuition and "matching integrands" is rigorous enough. I initially tried to use Cauchy sums to prove this result, but I got stuck after realizing that Cauchy sums are primarily used to prove that functions are Riemann integrable, whereas in this question, we want to prove that the two functions $f$ and $g$ are equivalent. Any guidance would be appreciated!

Answer 1

Note that $\int_c^{d} f(x)dx=\int_c^{d} g(x)dx$ if $a \leq c <d \leq b$. Suppose $f(s) >g(s)$ for some $s$. Let $r=f(s)-g(s)$. Then, by continuity, we can find an interval $(c,d)$ such that $f(t)>g(t)+\frac r 2$ for all $t$ in $(c,d)$. This implies that $\int_c^{d} f(x)dx> \int_c^{d} g(x)dx +\frac {r(d-c)} 2$, a contradiction.